Degree of Parabolic Quantum Groups - Dipartimento di Matematica ...

5. Quantum universal enveloping algebras for parabolic Lie algebras 59
Proposition 5.4.2. For t = 1, we have U 1 ǫ = Uǫ
Proof. For t = 1, the relations of U 1 ǫ are exactly the relation 4.2.2 that define
Uǫ.
Let 0 = λ ∈ C, define
ϑλ(Ei) = 1
λ Ei, ϑλ(Fi) = 1
λ Fi, ϑλ(Li) = 1
λ Li, ϑλ(K ±1
i ) = K ±1
i , (5.10)
for i = 1, . . .,n.
Proposition 5.4.3. For any 0 = λ ∈ C, ϑλ is an isomorphism of algebra
between U t ǫ and U λt
ǫ
Proof. Simple verification of the property.
Set Sǫ := U t=0
ǫ , we want to construct an explicit realization of it. Let
D = Uǫ(b+) ⊗ Uǫ(b−) and define the map
Σ : Sǫ → D
by Σ(Ei) = Ei := Ei ⊗ 1, Σ(Fi) = Fi = 1 ⊗ Fi, and Σ(K ±1
i ) = K ±1
K ±1
i ⊗ K±1
i for i = 1, . . .,n.
Lemma 5.4.4. Σ is a well defined map.
i :=
Proof. We must verify that the image of Ei, Fi and Ki satisfy the relation
5.6 for t = 0.
±
K i K ± j = K± j K± i
(5.11)
KiK −1
= 1
i
KiEjK −1
i = ǫaijEj KiFjK −1
i = ǫ−aijFj ⎧
⎪⎨
⎪⎩
[Ei, Fi] = 0
1−aij r=0 (−1)r
1 − aij
1−aij
r=0 (−1)r
r
1 − aij
r
ǫi
ǫi
E 1−aij−r
i EjE r i
F 1−aij−r
i FjF r i
= 0
= 0
(5.12)
(5.13)
Note that the relations 5.11 are obvious. We begin by demonstrating the
relation 5.12
KiEjK −1
i
= Ki ⊗ Ki (Ei ⊗ 1) K −1
i
= KiEiK −1
i
= ǫ aij Ei ⊗ 1
= ǫ aij Ei
⊗ 1
⊗ K−1
i